Global Existence for Rate-Independent Gradient Plasticity

Transcription

1 Global Existence for Rate-Independent Gradient Plasticity Alexander Mielke Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin Institut für Mathematik, Humboldt-Universität zu Berlin CRM De Giorgi, Pisa, 17 November 2007 Rate-Independence, Homogenization and Multiscaling Joint work with Andreas Mainik with credits to Stefan Müller and Gilles Francfort

2 1. Introduction Overview 1. Introduction 2. Finite-strain elastoplasticity 3. Energetic solutions for rate-independent systems 4. Existence results 5. Time-dependent boundary data A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 2/24

3 1. Introduction Finite-strain relates to geometric nonlinearities multiplicative decomposition ϕ = F = F elast F plast plastic yielding gives nonsmoothness partially hysteretic behavior in loading cycling ϕ ϕ(ω) Ω ϕ(x) x Develop sufficient conditions for existence such as strong hardening gradient regularization to understand the mechanisms for failure localization, shearbands microstructure formation A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 3/24

4 1. Introduction Finite-strain relates to geometric nonlinearities invariance under rigid-body transformations (frame indifference) C = F T F symmetric Cauchy strain tensor no self-penetration, i.e., det F > 0 multiplicative decomposition F = F elast P or F elast = ϕ P 1 F GL + (d) := { A R d d det A > 0 } and P SL(d) := { A R d d det A = 1 } Use (sub) Riemannian and Finslerian geometry on the Lie groups GL + (d), SL(d),... Nonsmoothness is treated by a weak solution concept: energetic solutions do not need derivatives A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 4/24

5 1. Introduction Literature Ortiz et al 1999,..., Miehe et al 2002,... M/Theil/Levitas 1999, NoDEA 2001/04, ARMA 2002 Carstensen/Hackl/M [PRSL 2002] Non-convex potentials and microstructure in finite-strain plasticity M. [CMT 2003] Energetic formulation of multiplicative elastoplasticity M. [SIMA 2004] Existence of minimizers in incremental elastoplasticity Francfort/M. [JRAM 2006] Existence results for rate-independent material models. (based on Dal Maso et al) M/Müller [ZAMM 2006] Lower semicontinuity and existence of minimizers for a functional in elastoplasticity Mainik/M Global existence for rate-independent strain-gradient plasticity at finite strains. In preparation. A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 5/24

6 2. Finite-strain elastoplasticity Overview 1. Introduction 2. Finite-strain elastoplasticity 3. Energetic solutions for rate-independent systems 4. Existence results 5. Time-dependent boundary data A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 6/24

7 2. Finite-strain elastoplasticity Ω x ϕ F= ϕ P=F plast F elast ϕ(ω) ϕ(x) P plastic tensor p hardening variables Multiplicative decomposition ϕ = F = F elast F plast P = F plast SL(d), F elast = ϕ P 1 For elastic properties only F elast matters. Plastic invariance (except for hardening) R(P, p, Ṗ, ṗ) = R(ṖP 1, p, ṗ) Energy-storage functional E(t, ϕ, P, p) = Ω W ( ϕp 1 ) + H(P, p) + U(P, p, P, p) }{{}}{{}}{{} elastic energy hardening regularization dx l(t),ϕ Elastic equilibrium: 0 = D ϕ E(t,ϕ, P, p) Plastic flow law: 0 ṖR(P, p, Ṗ, ṗ) + D PE(t,ϕ, P, p) 0 ṗr(p, p, Ṗ, ṗ) + D pe(t,ϕ, P, p) A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 7/24

8 2. Finite-strain elastoplasticity Rayleigh s dissipation potential R(P, p, Ṗ, ṗ) Rate independence R(P, p,γṗ,γṗ) = γ1 R(P, p, Ṗ, ṗ) for γ 0. (sub) Riemannian/Finslerian metric on SL(d) R m. To avoid rates we introduce a dissipation distance { 1 D(P 0, p 0, P 1, p 1 ) = min 0 R( P, p, P, p)ds ( P, p)(0))=(p 0, p 0 ), } ( P, p)(1))=(p 1, p 1 ), ( P, p) C 1 ([0, 1]; SL(d) R m ) Triangle inequality D(P 0, p 0, P 2, p 2 ) D(P 0, p 0, P 1, p 1 ) + D(P 1, p 1, P 2, p 2 ) No symmetry, in general D(P 0, p 0, P 1, p 1 ) D(P 1, p 1, P 0, p 0 ) Plastic invariance D(P 0 P, p 0, P 1 P, p 1 ) = D(P 0, p 0, P 1, p 1 ) for all P SL(d) A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 8/24

9 2. Finite-strain elastoplasticity D(P 0 P, p 0, P 1 P, p 1 ) = D(P 0, p 0, P 1, p 1 ) for all P SL(d) Plastic invariance implies D(P 0, p 0, P 1, p 1 ) = D(I, p 0, P 1 P 1 0, p 1) D(I, p 0, P, p 1 ) log P nonconvex, bad coercivity von Mises plasticity: isotropic hardening p [0, ) { A R(P, p, Ṗ, ṗ) = (p)ṗ if ṖP 1 vm ṗ, else, with ξ vm = ( α ξ+ξ T 2 + β ξ ξ T 2) 1/2 β = 0: D(I, p 0, P, p 1 ) = A(p 1 ) A(p 0 ) for p 1 p 0 α 2 β > 0: No formula is known! Under what conditions is P D(I, p 0, P, p 1 ) polyconvex? log(p T P) vm A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 9/24

10 3. Energetic solutions for rate-independent systems Overview 1. Introduction 2. Finite-strain elastoplasticity 3. Energetic solutions for rate-independent systems 4. Existence results 5. Time-dependent boundary data A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 10/24

11 3. Energetic solutions for rate-independent systems Admissible deformations Γ Dir Ω F = {ϕ W 1,q ϕ (Ω; R d ) ϕ = ϕ Dir on Γ Dir } (for the moment, we assume time-independent Dirichlet data) Set of internal states z = (P, p) Z = {(P,p) W 1,r P(Ω;R d ) W 1,r p (Ω;R m ) P SL(d) a.e. in Ω } Energy-storage functional (as above) E : [0, T] F Z R := R { } (t,ϕ, z) Ω W ( ϕp 1 ) + H(z) + U(z, z)dx l(t),ϕ Dissipation distance on Z: D : Z Z [0, ] (z 0, z 1 ) = (P 0, p 0, P 1, p 1 ) Ω D(P 0(x), p 0 (x), P 1 (x), p 1 (x))dx. A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 11/24

12 3. Energetic solutions for rate-independent systems Definition [M/Theil 99, M. 03]: (ϕ, z) : [0, T] F Z is called an energetic solution associated with E and D, if for all t [0, T] the stability condition (S) and the energy balance (E) hold: (S) ( ϕ, z) F Z: E(t,ϕ(t), z(t)) E(t, ϕ, z) + D(z(t), z) (E) E(t,ϕ(t), z(t)) + Diss D (z,[0, t]) = E(0,ϕ(0), z(0)) + t 0 τ E(τ,ϕ(τ), z(τ)) dτ }{{} power ext. forces Any sufficiently smooth energetic solution solves Elastic equilibrium: 0 = D ϕ E(t,ϕ, z) Plastic flow law: 0 żr(z, ż) + D z E(t,ϕ, z) However, in general energetic solutions have jumps w.r.t. to time. A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 12/24

13 4. Existence results Overview 1. Introduction 2. Finite-strain elastoplasticity 3. Energetic solutions for rate-independent systems 4. Existence results 5. Time-dependent boundary data A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 13/24

14 4. Existence results Main assumptions on E and D: E(t, ϕ, P, p) Ω W( ϕp 1 )+H(P, p)+u(p, p, P, p)dx l(t), ϕ D(P 0, p 0, P 1, p 1 ) = Ω D(P 0, p 0, P 1, p 1 )dx (A1) Lower semicontinuity: W : R d d R is polyconvex and lsc H : SL(d) R m R is lsc U continuous and U(P, p,, ) is convex D : (SL(d) R m ) (SL(d) R m ) [0, ] lsc (A2) Coercivity: W (F el ) c F el q F C H(P, p) + D(I, p, P, p) c ( P q P+ P 1 q P+ p r ) p C U(P, p, P, p) c ( P r P+ p r ) p C A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 14/24

15 4. Existence results Proposition. Let (A1) and (A2) hold with d q P + 1 q F < 1 and r p, r P > 1. Define q ϕ via 1 q ϕ = 1 q P + 1 q F and assume q ϕ > d and l C 1( [0, T],(W 1,q ϕ (Ω; R d )) ). Consider F W 1,q ϕ (Ω; R d ) and Z ( W 1,r P(Ω; R d d ) L q P(Ω; SL(d)) ) W 1,r p (Ω; R m ) equipped with the weak topologies, respectively. Then, D is weakly lower semicontinuous on Z Z E(t, ) has weakly compact sublevels in [0, T] F Z ( coercivity and lower semicontinuity) Lower semicontinuity ϕ k ϕ in L q ϕ and P k P in L q P = minors converge: M s ( ϕ k P 1 k ) M s( ϕp 1 ) in L q s A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 15/24

16 4. Existence results Existence via the Incremental Minimization Problem (IP): Partitions 0 = t0 N < tn 1 < < tn K N 1 < tn K N = T (ϕ N k, zn k ) Arg Min{ E(tk N, ϕ, z) + D(zN k 1, z) ϕ F, z Z }. Piecewise constant interpolants (ϕ N, z N ) : [0, T] F Z. Limit passage needs the Joint Recovery Condition If (ϕ n, z n ) stable at t n, (t n,ϕ n, z n ) (t,ϕ, z), E(t n,ϕ n, z n ) C and ( ϕ, ẑ) F Z, then there exists ( ϕ n, ẑ n ) such that lim sup E(t n, ϕ n, ẑ n ) + D(z n, ẑ n ) E(t n,ϕ n, z n ) n E(t, ϕ, ẑ) + D(z, ẑ) E(t,ϕ, z) Easy to satisfy, if D is weakly continuous: ẑ n = ẑ A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 16/24

17 4. Existence results Existence Result [Mainik/M. 07] Assume (A1), (A2) and that either (i) or (ii) holds: (i) D : (SL(d) R m ) (SL(d) R m ) [0, ) is continuous and D(P 0, p 0, P 1, z 1 ) C ( P 0 q P+ P 1 q P+ p 0 r p + p 1 r p ) (ii) r P, r p > d and for each (P 0, p 0 ) the set {(P 1, p 1 ) D(P 0, p 0, P 1, p 1 ) < } has nonempty interior. Then, for each stable initial state (ϕ 0, z 0 ) with D(I, p, z 0 ) < there exists an energetic solution (ϕ, z) : [0, T] F Z. Case (i) applies for kinematic hardening (no p needed) Case (ii) is needed for isotropic hardening, i.e., p R: H(P, p) = exp(κp), D(I, p 0, P, p 1 ) = { p1 p 0 for p 1 p 0 log(p T P) else A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 17/24

18 5. Time-dependent boundary data Overview 1. Introduction 2. Finite-strain elastoplasticity 3. Energetic solutions for rate-independent systems 4. Existence results 5. Time-dependent boundary data A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 18/24

19 5. Time-dependent boundary data Essential idea in Francfort/M using Francfort, Dal Maso, Toader 2005 and Ball 2002 Dirchlet data ϕ(t, x) = ϕ Dir (t, x) for (t, x) [0, T] Γ Dir. Usual additive ansatz: ϕ(t, ) = ϕ Dir (t, ) + ψ(t, ) with ψ(t) F = {ψ W 1,q ϕ (Ω; R d ) ψ ΓDir = 0 } E(t,ψ, z) = E(t,ϕ Dir (t)+ψ, z) Power of external loading (now including power of ϕ Dir ) t E(t,ψ, z) = Ω F W (( ϕ Dir (t)+ ψ)p 1 ): ( ϕ }{{} Dir P 1) }{{} 1 st Piola-Kirchhoff tensor.t L q (Ω) dx l(t),ψ In general we cannot expect T L q (Ω), not even L 1 (Ω) A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 19/24

20 5. Time-dependent boundary data Way out? Matrices are elements of Lie groups: Multiply!! Extend ϕ Dir to all of R d, namely g Dir C 2 ([0, T] R d ; R d ) such that g Dir,( g Dir ) 1 L ([0, T] R d ; R d d ) ϕ(t, x) = g Dir (t,ψ(t, x)) with ψ(t) F = {ψ W 1,q ϕ (Ω; R d ) ψ ΓDir = id ΓDir } Chain rule replaces linearity: x ϕ(t, x) = y g Dir (t,ψ(t, x)) x ψ(t, x) Ẽ(t,ψ, z) = E(t, g Dir (t, ) ψ, z) Power of external loading t Ẽ(t,ψ, z) = Ω F W ( g Dir (t,ψ) ψp 1 ): ( ġ Dir (t,ψ) ψp 1) dx A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 20/24

21 5. Time-dependent boundary data t Ẽ(t,ψ, z) = = Ω Ω F W ( g }{{ Dir ψ P 1 ): ( ġ }}{{} Dir ψp 1) dx }{{} G F H F W (GFP 1 )(GFP 1 ) T : ( HFP 1 (GFP 1 ) 1) }{{}}{{} Kirchhoff K(GFP 1 ) V:=HG 1 dx V L ([0, T] Ω; R d d ) independently of ψ : [0, T] F Multiplicative stress control (Baumann,Phillips,Owen 91, Ball 02) Kirchhoff tensor K(F) := F W (F)F T c 0, c 1 F GL + (d) + (R d ) : K(F) c 1 ( W (F)+c0 ) A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 21/24

22 5. Time-dependent boundary data Multiplicative stress control is compatible with finite-strain polyconvexity: W (F) = α F q + β (det F) γ T(F) = αq F q 2 F K = TF T = αq F q 2 FF T βγ (det F) γ+1 cof F (not bounded by W ) βγ (det F) γ I Hence, K(F) maxα,β dw (F). (as (cof F) F T = (det F)I) Corollary: There exist c0 E, ce 1 such that t Ẽ(t,ψ, z) c1 E (Ẽ(t,ψ, ) z) + c E 0 whenever the right-hande side is finite. Provides useful a priori energetic estimates needs of course sufficient hardening Alber s group: Self-controling property A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 22/24

23 Conclusion Using full regularization and strong coercivity the existence of energetic solutions can be shown for many plasticity models. Future tasks: Study formation of microstructure for vanishing regularization: U(z, z) = ε z r with ε 0 Study failure via localization for vanishing strong coercivity H(P, p) = p 2 + ε exp(κ p ) for ε 0 deformation gradients ϕ ε behaves badly, but energy densities and stresses have suitable limits (cf. [Bouchitté/M/Roubíček 07]) Replace global minimization by local minimization in (IP) viscous approximations A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 23/24

24 Thank you for your attention! Papers available under C. Carstensen, K. Hackl, and A. Mielke. Non-convex potentials and microstructures in finite-strain plasticity. Proc. Royal Soc. London, Ser. A, 458: , A. Mielke. Existence of minimizers in incremental elasto-plasticity with finite strains. SIAM J. Math. Analysis, 36: , A. Mielke and S. Müller. Lower semicontinuity and existence of minimizers for a functional in elastoplasticity. Z. angew. Math. Mech., 86: , E. Gürses, A. Mainik, C. Miehe, and A. Mielke. Analytical and numerical methods for finite-strain elastoplasticity. In Helmig/Mielke/Wohlmuth (eds), Multifield problems in Fluid and Solid Mechanics. Springer 2006, pp (WIAS preprint 1127). A. Mainik, A. Mielke. Global existence for rate-independent gradient plasticity at finite strain. December A. Mielke Pisa, 17 Nov. 2007, Rate-Indep., Homogenization and Multiscaling 24/24